Math Geek Mom: The Normal Distribution
Imagine a number line, extending in both directions infinitely. Above this line we might graph bars that represent the proportion of observations of something that fall within any given interval on the number line. We can do this for much of the data sets that show up in nature, such as the length of a leaf on a tree, the height of a grown woman, the average body temperature or even the length of a human life. When we start graphing these data that show up in nature, we notice that they tend to all look slightly similar.
Imagine a number line, extending in both directions infinitely. Above this line we might graph bars that represent the proportion of observations of something that fall within any given interval on the number line. We can do this for much of the data sets that show up in nature, such as the length of a leaf on a tree, the height of a grown woman, the average body temperature or even the length of a human life. When we start graphing these data that show up in nature, we notice that they tend to all look slightly similar. In each case, the mean value is in the center, with a declining percentage of observations occurring as one moves away from the mean, in either the positive or negative direction. If we smooth out these graphs, so that we are no longer graphing over set intervals on the number line, but drawing the graph over the entire number line, we find something called the “normal distribution”. Some people call this the “bell curve” since it looks like an upside down bell that may be either concentrated closely at the mean or spread out widely on either side from that mean. The distance of any point from that mean can be measured in “standard deviations” that are large if the curve is flat and wide or small if the curve is tall and thin. I thought of this recently as I listened to my friend “Jan” tell me the latest chapter in the story of her daughter.
“Jan” is a breast cancer survivor who adopted a beautiful little girl named “Lynn” after beating, we assume, the disease about ten years ago. Lynn has large brown eyes and a head of amazing brown curly hair that falls to her shoulders. Since the time she was a baby, Lynn would look around at the world and seem to take it all in, her eyes sparkling with recognition at what she saw. We would always say that we could not wait to hear what she had to say once she started talking. And once she did, it was interesting to hear how her mind worked. She made interesting connections between things and helped us all see the world in a different way. I remember hearing that one year, when Jan and her family were going on vacation, Lynn, who was in preschool at the time, heard that they were going to stay at a hotel. When she asked about it, she said “when are we getting to the ‘show and tell’”? And, more recently, when she was describing the house across the street where almost constant NBA playoff parties were being held, she called the house “the Cavs house”, since this is Cleveland, and most of the partiers were attending in attire supporting the Cavaliers, the local NBA team. She notices patterns everywhere, often telling us what the next piece of a pattern would be, whether it be the color of the next house or the next number in a telephone number. We all adore her and think that her mind, that sees things a little differently than most people’s, will make an important contribution to the world someday.
So we were all shocked when the IQ testing her school did came back recently. It seems that she tested at between “low normal” to the upper end of “below normal”. I found myself reminding Jan that even Einstein didn’t excel in mathematics, which was, of course, the basis of his work as a physicist. I have to wonder if some of the results on this test are because Lynn sees the world in such a unique way. When asked to choose the object that did not belong to a set of objects, she frequently chose the wrong one. When Jan asked whether they asked her why she chose the one she did, they said they did not, as they were just looking to see if they chose the correct answer. I wondered what that beautiful little girl might tell us about the set of pictures if she had a chance to explain herself.
When I teach the normal distribution in my statistics classes, I often use the IQ test as an example. It is very elegant, with a mean of 100 and a standard deviation of 15, giving nice, even results that students don’t stumble over. I always begin my lecture on it by saying that what the IQ test does is test the person’s ability to take that particular test on that particular day.
I must confess, however, that I say this with a grain of smugness in my voice. I do so because I remember that, years ago, in high school, I took such a test and obtained a very high score. Of course, that was before my brain was cut open to remove something the size of an orange that had no business being in it. I have no idea what I would score today.
I do know one thing, however. As I talk to my friend’s brilliant child who somehow did not do well on their standardized test, I know that that minor air of smugness will be gone from my voice forever. Now I REALLY believe that these tests are not a good measure of true intelligence.
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